Related papers: Cellular Algebras and Graph Invariants Based on Qu…
We give a short overview over recent developments on quantum graphs and outline the connection between general quantum graphs and so-called quantum random walks.
Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear…
Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the case of dynamic graphs, in which an…
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise quantum walks have shown much potential as a frame- work for developing new quantum algorithms.…
Currently there are three major paradigms of quantum computation, the gate model, annealing, and walks on graphs. The gate model and quantum walks on graphs are universal computation models, while annealing plays within a specific subset of…
Non Commutative Geometry (NCG) is considered in the context of a charged particle moving in a uniform magnetic field. The classical and quantum mechanical treatments are revisited and a new marker of NCG is introduced. This marker is then…
We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW) and Quantum Cellular Automata (QCA), in (1 + 1)-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit…
In this report, we describe a novel graph invariant for computational graphs (colored directed acylic graphs) and how we used it to generate all distinct computational graphs up to isomorphism for small graphs. The algorithm iteratively…
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which…
A quantum walk is a time-homogeneous quantum-mechanical process on a graph defined by analogy to classical random walk. The quantum walker is a particle that moves from a given vertex to adjacent vertices in quantum superposition. Here we…
Evolution algebras are a special class of non-associative algebras exhibiting connections with different fields of Mathematics. Hilbert evolution algebras generalize the concept through a framework of Hilbert spaces. This allows to deal…
Quantum random walks have been shown to be powerful quantum algorithms for certain tasks on graphs like database searching, quantum simulations etc. In this work we focus on its applications for the graph isomorphism problem. In particular…
If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights…
There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained…
In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $\Gamma$ and a vertex-transitive subgroup $G$ of $\mathrm{Aut}(\Gamma)$, a $2$-factor $\mathcal{C}$…
We consider quantum walks on a finite graphs to which infinite tails are attached. We explore how the propagating and bound states depend on the structure of the finite graph. The S-matrix for such graphs is defined. Its unitarity is proved…
A cellular algebra is called cyclic cellular if all cell modules are cyclic. Most important examples of cellular algebras appearing in representation theory are in fact cyclic cellular. We prove that if $A$ is a cyclic cellular algebra,…
Owing to their versatility, graph structures admit representations of intricate relationships between the separate entities comprising the data. We formalise the notion of connection between two vertex sets in terms of edge and vertex…
Originating from spectral graph theory, cospectrality is a powerful generalization of exchange symmetry and can be applied to all real-valued symmetric matrices. Two vertices of an undirected graph with real edge weights are cospectral iff…
We introduce quantum walks on Cayley graphs of non-Abelian groups. We focus on the easiest case of virtually Abelian groups, and introduce a technique to reduce the quantum walk to an equivalent one on an Abelian group with coin system…